I have a wire that stretches from $x=0$ to $\infty$. The temperature at $x=0$ is given by the unknown function $f(t)$ for $t$ from $-\infty$ to now ($t=0$).
I can measure the temperature of the wire at each point now ($t=0$), $g(x)$.
Given the temperature of the wire now, $g(x)$, I would like to recover $f(t)$, the temperature at $x=0$, through history
The problem is likely "ill posed" meaning tiny errors in $g(x)$ lead to big errors in $f(t)$.
What is known about solving this problem? For example, is the degree of ill-posedness known? If it is mildly ill-posed, are there numerical techniques available? Can anyone point me to articles or a book that treats it?